Not really. That is just a fact that there's only 365 days, and the more samples you make increases the odds it's a sample that overlaps with another (there are fewer unique options).

What the OP is saying is that sometimes randomness can appear less random than other randomness. True randomness will occasionally give results that closely match something non-random. It's why almost all music players don't use true random for shuffle. True random you could have the same song play 15 times in a row. In fact, that is expected to happen eventually (assuming infinite time) just as all other sets of 15 songs are.